142. Zero Indices and Powers of Powers

Learning Intentions

Write each expression in expanded form, then simplify:
a) $(b^{4})^{2}$
b) $(x^{3})^{2}$
c) $(m^{2})^{3}$

Simplify each expression with a zero index:
a) $a^{0}$
b) $7 x^{0}$
c) $(p q)^{0}$

Simplify each power of a power:
a) $(y^{5})^{2}$
b) $(a^{2})^{4}$
c) $(p^{3})^{3}$

Expand each product taken to a power:
a) $(a b)^{2}$
b) $(x y)^{3}$
c) $(2 m)^{2}$

Expand and simplify:
a) $(a b)^{3}$
b) $(3 x)^{2}$
c) $(2 p q)^{3}$

Simplify each expression:
a) $(x^{2})^{3} \times x^{0}$
b) $(a b)^{2} \times (a b)$
c) $(m^{4})^{2} \div m^{3}$

Write each expression in expanded form, then simplify:
a) $(c^{3})^{2}$
b) $(y^{2})^{3}$
c) $(n^{4})^{2}$

Simplify each expression with a zero index:
a) $b^{0}$
b) $9 p^{0}$
c) $(r s)^{0}$

Simplify each power of a power:
a) $(z^{4})^{2}$
b) $(b^{3})^{4}$
c) $(q^{2})^{5}$

Expand each product taken to a power:
a) $(c d)^{2}$
b) $(m n)^{3}$
c) $(4 t)^{2}$

Expand and simplify:
a) $(x y)^{3}$
b) $(5 a)^{2}$
c) $(3 r s)^{3}$

Simplify each expression:
a) $(y^{3})^{2} \times y^{0}$
b) $(c d)^{2} \times (c d)$
c) $(p^{5})^{2} \div p^{4}$

Write each expression in expanded form:
a) $(x^{2})^{2}$
b) $(a^{3})^{2}$
c) $(m^{2})^{4}$
Simplify each power of a power:
a) $(x^{4})^{2}$
b) $(b^{3})^{3}$
c) $(p^{2})^{5}$
d) $(y^{6})^{2}$
Simplify each expression with a zero index:
a) $x^{0}$
b) $a^{0}$
c) $(m n)^{0}$
d) $5 p^{0}$
Simplify each expression:
a) $(q^{2})^{3}$
b) $(r^{5})^{2}$
c) $(t^{3})^{4}$
d) $(k^{1})^{3}$
Expand each product taken to a power:
a) $(a b)^{2}$
b) $(x y)^{3}$
c) $(2 m)^{2}$
d) $(3 p q)^{2}$
Expand and simplify each expression:
a) $(c d)^{3}$
b) $(4 x)^{2}$
c) $(2 a b)^{3}$
d) $(5 m n)^{2}$
Simplify each expression:
a) $(x^{2})^{3} \times x^{0}$
b) $(a b)^{2} \times a b$
c) $(m^{4})^{2} \div m^{5}$
d) $(2 p)^{2} \times p^{3}$
Simplify each expression:
a) $(y^{3})^{2} \div y$
b) $(r s)^{2} \times (r s)^{2}$
c) $(a^{5})^{2} \times a^{0}$
d) $(3 x)^{2} \div x$

Explain why $(b^{4})^{2}$ means $(b^{4}) \times (b^{4})$ .
A student says that $x^{0} = 0$ . Explain the mistake.
Noah says that $(a^{3})^{2} = a^{5}$ because $3 + 2 = 5$ . Is he correct? Explain.
Explain why $(a b)^{3} = a^{3} b^{3}$ .
A student expands $(2 x)^{2}$ as $2 x^{2}$ . Describe the error.

A pattern rule includes $(m^{2})^{3}$ . Write this in simplest form.
A model for area uses $(3 x)^{2}$ . Expand and simplify the expression.
A science formula contains $(a b)^{3} \times a b$ . Simplify the result.
A computer process gives $(p^{4})^{2} \div p^{3}$ . Simplify the output.
A student writes a term as $(x y)^{2} \times (x y)^{3}$ . Simplify the term.
A design uses $(2 m n)^{2}$ tiles in one section. Expand and simplify this expression.

Students may think $(b^{4})^{2}$ means $b^{4 + 2}$ instead of multiplying the indices
Students may think a zero index gives the value $0$ instead of $1$
Students may forget that the zero index rule applies only when the base is not zero
Students may confuse a power of a power with multiplying terms that have the same base
Students may add indices instead of multiplying them in expressions like $(x^{3})^{2}$
Students may think $(a b)^{3}$ means $a b^{3}$ rather than $a^{3} b^{3}$
Students may forget to apply the power to the numerical coefficient as well, for example in $(2 x)^{2}$
Students may simplify $(3 x)^{2}$ as $3 x^{2}$ instead of $9 x^{2}$
Students may not use brackets carefully and lose track of what the power applies to